2020/11/16 by Ballantine, Cristina, Welch, Amanda
#05A17 #11P81 #11P83 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2011.08220
Let \mathcal Or(n) be the set of r-regular partitions of n, \mathcal Dr(n) the set of partitions of n with parts repeated at most r-1 times, \mathcal O1,r(n) the set of partitions with exactly one part (possibly repeated) divisible by r, and let \mathcal D1,r(n) be the set of partitions in which exactly one part appears at least r times. If Er, t(n) is the excess in the number of parts congruent to t \pmod r in all partitions in \mathcal Or(n) over the number of different parts appearing at least t times in all partitions in \mathcal Dr(n), then Er, t(n) = |\mathcal O1,r(n)| = |\mathcal D1,r(n)|. We prove this analytically and combinatorially using a bijection due to Xiong and Keith. As a corollary, we obtain the first Beck-type identity, i.e., the excess in the number of parts in all partitions in Or(n) over the number of parts in all partitions in Dr(n) equals (r - 1)|O1,r(n)| and also (r - 1)|D1,r(n)|. Our work provides a new combinatorial proof of this result that does not use Glaisher's bijection. We also give a new combinatorial proof based of the Xiong-Keith bijection for a second Beck-Type identity that has been proved previously using Glaisher's bijection.